MILAN, Dec 29 (Reuters) - Italian bank bailout fund Atlante has presented an offer for two-thirds of the 3.7 billion euros ($4 billion) of gross problematic loans of three small banks that were rescued last year, two sources close to the matter said.

The move should help facilitate the sale of the banks - Banca Etruria, Banca Marche and CariChieti - to bigger rival and Italy’s fifth-largest lender UBI.

“Atlante today presented an offer for the non-performing loans of the three good banks,” one of the sources said.

A third source added that the banks’ boards may meet on Friday to discuss the Atlante deal.

In November 2015, Italy rescued the three small lenders and a fourth, CariFerrara, by drawing nearly 4 billion euros from a crisis fund paid for by its healthy banks. The value of junior bonds and shares in the banks was written off in line with European rules for dealing with bank crises.

Italy has struggled to sell the four banks as required by European regulators even though their weakest non-performing loans were spun off in the rescue. Since then the banks have been forced to class more problem loans as defaulting.

The small lenders have turned into the latest banking headache for Italy’s government, which has already been called to the rescue of the world’s oldest bank, Monte dei Paschi di Siena, after its plan to raise 5 billion euros from private investors flopped.

UBI has expressed interest in buying three of the lenders, but set conditions including for the banks’ new non-performing loans to be taken off their balance sheets and the option to use its own internal risk models to weigh the lenders’ assets.

The resolution fund which owns the banks is also expected to carry out a small share sale to bolster their capital ratios. Once all conditions are fulfilled, UBI would launch its own small cash call to ensure its core capital is not hit by the acquisition.

The sale to UBI is unlikely to be completed by the end of the year as had initially been expected.

“There are still legal issues to be ironed out, so UBI’s signature is expected after the Epiphany (Jan. 6 holiday),” one of the sources said.

UBI, whose board was meeting on the matter on Thursday, declined to comment.

When the main banking unions met Roberto Nicastro, who acts as president of the four banks until they are sold, in early December, he expected a preliminary sale agreement to be signed this year, with closing expected by March 2017.

The deadline for the sale, initially set for April 2016, has already been extended twice. (Reporting by Maria Pia Quaglia, writing by Agnieszka Flak; Editing by Ruth Pitchford)